No, for JSmol you don't need Java any more. JavaScript is not Java (Oracle), but still is run on client side. Although JavaScript doesn't require any plugins.

Just to cite header of Jmol2.js file, which Bob (Hanson) provided:

Script replacement for legacy Jmol.js that uses JSmol instead.
Can be used to turn most legacy Jmol.js-based sites to JSmol.
Presumes prior loading of JSmol.min.js
1) rename your current Jmol.js file Jmol_old.js in case you want to undo this
2) replace that file with this one
3) If you use JmolInitialize, put the Jar files from the java/ directory here
in that designated directory and make a new subdirectory there called j2s.
Then put all j2s/* files in that j2s subdirectory.
If you don't use JmolIinitialize, all the jar files should go into a subdirectory named "java"
in the same directory as your web page, and all the JavaScript should go into a subdirectory "j2s"
also in the same directory as your web page, just like it is here.
3) copy all j2s/* files into a directory on your site if you want to use HTML5 (defaults to ./j2s)
4) copy all java/* files into a directory on your site if you want to use Java (defaults to ./java)
5) try your page and see how it goes. You may still have some problems, because not all of the
methods in the original Jmol.js are included here. Let me know if that's the case.

OK, I read a little further down in the Wiki and saw that I could accomplish the switch another way: leave the calls to jmolApplet(), etc. alone and instead use both JSmol.min.js and Jmol2.js as resources. So I did so, and I also added a symlink to the
jsmol/j2s folder in the same folder as the page that calls the "applet". So now JSmol loads, but I get this alert:

jmolLoadInline not implemented

So, any chance of this method being implemented in the near future? Or maybe it has already been implemented in a more recent version? I'm using jsmol-13.3.4. (BTW, I tried to download JSmol 13.3.5 from SourceForge, but decompression failed. Instead
I downloaded one of the 13.3.4 versions.)